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Theorem difdif 4130
Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
difdif (𝐴 ∖ (𝐵𝐴)) = 𝐴

Proof of Theorem difdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm4.45im 826 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)))
2 iman 400 . . . . 5 ((𝑥𝐵𝑥𝐴) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
3 eldif 3957 . . . . 5 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
42, 3xchbinxr 334 . . . 4 ((𝑥𝐵𝑥𝐴) ↔ ¬ 𝑥 ∈ (𝐵𝐴))
54anbi2i 621 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
61, 5bitr2i 275 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ 𝑥𝐴)
76difeqri 4123 1 (𝐴 ∖ (𝐵𝐴)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wcel 2099  cdif 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-dif 3950
This theorem is referenced by:  dif0  4377  undifabs  4482
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